GATE EE
$${{{d^2}V} \over {d{t^2}}} + a.{{dV} \over {dt}} + bV = f\left( t \right).$$
Find $$a,b$$ and $$f(t)$$


$$(i)$$ Calculate the transfer function $${{{V_0}} \over {{V_i}}}$$
$$(ii)$$ plot the amplitude and phase response as a function for $$R = {R_1}$$

$$(i)$$ Draw the $$AC$$ equivalent circuit
$$(ii)$$ Find the voltage gain of the amp


The Nyquist plot of $$G$$ encircle the origin






$$(a)$$ Constant excitation and non-zero leading power-factor
$$(b)$$ Constant excitation and zero power-factor, leading
$$(c)$$ Constant terminal voltage and zero power-factor, leading
$$(d)$$ Constant terminal voltage and non-zero leading power-factor

$$e\left( t \right) = \sqrt 2 .00\,\,\sin \,\,314t$$

Transfer functions
$$\eqalign{
& \left( a \right)\,\,\,\,\,\,\,\,{1 \over {s\left( {s + 1} \right)}} \cr
& \left( b \right)\,\,\,\,\,\,\,\,{1 \over {{{\left( {s + 1} \right)}^2}}} \cr
& \left( c \right)\,\,\,\,\,\,\,\,{1 \over {s\left( {s + 1} \right) + 1}} \cr
& \left( d \right)\,\,\,\,\,\,\,\,{1 \over {{s^2} + 1}} \cr} $$
Impulse Responses